
# 2: First Order Differential Equations


• 2.1: Difference Equations
Differential equation are great for modeling situations where there is a continually changing population or value. If the change happens incrementally rather than continuously then differential equations have their shortcomings. Instead we will use difference equations which are recursively defined sequences.
• 2.2: Classification of Differential Equations
Recall that a differential equation is an equation (has an equal sign) that involves derivatives. Just as biologists have a classification system for life, mathematicians have a classification system for differential equations. We can place all differential equation into two types: ordinary differential equation and partial differential equations.
• 2.3: Modeling with First Order Differential Equations
Whenever there is a process to be investigated, a mathematical model becomes a possibility. Since most processes involve something changing, derivatives come into play resulting in a differential equation. We will investigate examples of how differential equations can model such processes.
• 2.4: Separable Differential Equations
A differential equation is called separable if it can be written as f(y)dy=g(x)dx
• 2.5: Autonomous Differential Equations
A differential equation is called autonomous if it can be written as y'(t)=f(y). Autonomous differential equations are separable and can be solved by simple integration.
• 2.6: First Order Linear Differential Equations
In this section we will concentrate on first order linear differential equations. Recall that this means that only a first derivative appears in the differential equation and that the equation is linear.
• 2.7: Exact Differential Equations
That is if a differential equation can be written in a specific form, then we can seek the original function f(x,y) (called a potential function). A differential equation with a potential function is called exact. If you have had vector calculus, this is the same as finding the potential functions and using the fundamental theorem of line integrals.
• 2.8: Theory of Existence and Uniqueness
If a first order differential satisfies continuity conditions, then the initial value problem will have a unique solution in some neighborhood of the initial value.
• 2.9: Theory of Linear vs. Nonlinear Differential Equations
There exists a solution to all first order linear differential equations.